3.1747 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=424 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{2 e^7 (a+b x) (d+e x)^2}+\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}-\frac {10 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^6 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (-5 a B e-A b e+6 b B d)}{3 e^7 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e) (-2 a B e-A b e+3 b B d)}{2 e^7 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)} \]

[Out]

-10*b^2*(-a*e+b*d)^2*(-A*b*e-B*a*e+2*B*b*d)*x*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-1/2*(-a*e+b*d)^5*(-A*e+B*d)*((b*x+
a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^2+(-a*e+b*d)^4*(-5*A*b*e-B*a*e+6*B*b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)+
5/2*b^3*(-a*e+b*d)*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^2*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-1/3*b^4*(-A*b*e-5*B*a*e+6*
B*b*d)*(e*x+d)^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+1/4*b^5*B*(e*x+d)^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+5*b*(-a*e+b*d
)^3*(-2*A*b*e-B*a*e+3*B*b*d)*ln(e*x+d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)

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Rubi [A]  time = 0.48, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \[ -\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^3 (-5 a B e-A b e+6 b B d)}{3 e^7 (a+b x)}+\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e) (-2 a B e-A b e+3 b B d)}{2 e^7 (a+b x)}-\frac {10 b^2 x \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{e^6 (a+b x)}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{e^7 (a+b x) (d+e x)}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{2 e^7 (a+b x) (d+e x)^2}+\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)}{e^7 (a+b x)}+\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^4}{4 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(-10*b^2*(b*d - a*e)^2*(2*b*B*d - A*b*e - a*B*e)*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^6*(a + b*x)) - ((b*d - a*
e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)*(d + e*x)^2) + ((b*d - a*e)^4*(6*b*B*d - 5*A*
b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*(d + e*x)) + (5*b^3*(b*d - a*e)*(3*b*B*d - A*b*e -
2*a*B*e)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*e^7*(a + b*x)) - (b^4*(6*b*B*d - A*b*e - 5*a*B*e)*(d +
e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (b^5*B*(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4
*e^7*(a + b*x)) + (5*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[d + e*x])/(
e^7*(a + b*x))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^3} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6}-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^3}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^2}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e) (d+e x)}{e^6}+\frac {b^9 (-6 b B d+A b e+5 a B e) (d+e x)^2}{e^6}+\frac {b^{10} B (d+e x)^3}{e^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {10 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) x \sqrt {a^2+2 a b x+b^2 x^2}}{e^6 (a+b x)}-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x) (d+e x)^2}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)}+\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^7 (a+b x)}-\frac {b^4 (6 b B d-A b e-5 a B e) (d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac {b^5 B (d+e x)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x)}+\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^7 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 501, normalized size = 1.18 \[ \frac {\sqrt {(a+b x)^2} \left (-6 a^5 e^5 (A e+B (d+2 e x))-30 a^4 b e^4 (A e (d+2 e x)-B d (3 d+4 e x))+60 a^3 b^2 e^3 \left (A d e (3 d+4 e x)+B \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )\right )+60 a^2 b^3 e^2 \left (A e \left (-5 d^3-4 d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )+B \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )\right )+10 a b^4 e \left (3 A e \left (7 d^4+2 d^3 e x-11 d^2 e^2 x^2-4 d e^3 x^3+e^4 x^4\right )+B \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )\right )+60 b (d+e x)^2 (b d-a e)^3 \log (d+e x) (-a B e-2 A b e+3 b B d)+b^5 \left (2 A e \left (-27 d^5+6 d^4 e x+63 d^3 e^2 x^2+20 d^2 e^3 x^3-5 d e^4 x^4+2 e^5 x^5\right )+3 B \left (22 d^6-16 d^5 e x-68 d^4 e^2 x^2-20 d^3 e^3 x^3+5 d^2 e^4 x^4-2 d e^5 x^5+e^6 x^6\right )\right )\right )}{12 e^7 (a+b x) (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(Sqrt[(a + b*x)^2]*(-6*a^5*e^5*(A*e + B*(d + 2*e*x)) - 30*a^4*b*e^4*(A*e*(d + 2*e*x) - B*d*(3*d + 4*e*x)) + 60
*a^3*b^2*e^3*(A*d*e*(3*d + 4*e*x) + B*(-5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3)) + 60*a^2*b^3*e^2*(A*e*(-
5*d^3 - 4*d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3) + B*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4))
+ 10*a*b^4*e*(3*A*e*(7*d^4 + 2*d^3*e*x - 11*d^2*e^2*x^2 - 4*d*e^3*x^3 + e^4*x^4) + B*(-27*d^5 + 6*d^4*e*x + 6
3*d^3*e^2*x^2 + 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5)) + b^5*(2*A*e*(-27*d^5 + 6*d^4*e*x + 63*d^3*e^2*x^2
+ 20*d^2*e^3*x^3 - 5*d*e^4*x^4 + 2*e^5*x^5) + 3*B*(22*d^6 - 16*d^5*e*x - 68*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 5*d
^2*e^4*x^4 - 2*d*e^5*x^5 + e^6*x^6)) + 60*b*(b*d - a*e)^3*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^2*Log[d + e*x]
))/(12*e^7*(a + b*x)*(d + e*x)^2)

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fricas [B]  time = 0.66, size = 871, normalized size = 2.05 \[ \frac {3 \, B b^{5} e^{6} x^{6} + 66 \, B b^{5} d^{6} - 6 \, A a^{5} e^{6} - 54 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 300 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 90 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} - 6 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} - 2 \, {\left (3 \, B b^{5} d e^{5} - 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 5 \, {\left (3 \, B b^{5} d^{2} e^{4} - 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} - 20 \, {\left (3 \, B b^{5} d^{3} e^{3} - 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} - 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} - 6 \, {\left (34 \, B b^{5} d^{4} e^{2} - 21 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 55 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 40 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5}\right )} x^{2} - 12 \, {\left (4 \, B b^{5} d^{5} e - {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} - 5 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 20 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} - 10 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x + 60 \, {\left (3 \, B b^{5} d^{6} - 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} - 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + {\left (3 \, B b^{5} d^{4} e^{2} - 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} - 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 2 \, {\left (3 \, B b^{5} d^{5} e - 2 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 6 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} - 6 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/12*(3*B*b^5*e^6*x^6 + 66*B*b^5*d^6 - 6*A*a^5*e^6 - 54*(5*B*a*b^4 + A*b^5)*d^5*e + 210*(2*B*a^2*b^3 + A*a*b^4
)*d^4*e^2 - 300*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 90*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 - 6*(B*a^5 + 5*A*a^4*b)*d
*e^5 - 2*(3*B*b^5*d*e^5 - 2*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 5*(3*B*b^5*d^2*e^4 - 2*(5*B*a*b^4 + A*b^5)*d*e^5 +
6*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 - 20*(3*B*b^5*d^3*e^3 - 2*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 6*(2*B*a^2*b^3 + A*
a*b^4)*d*e^5 - 6*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 - 6*(34*B*b^5*d^4*e^2 - 21*(5*B*a*b^4 + A*b^5)*d^3*e^3 + 55*
(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 40*(B*a^3*b^2 + A*a^2*b^3)*d*e^5)*x^2 - 12*(4*B*b^5*d^5*e - (5*B*a*b^4 + A*b
^5)*d^4*e^2 - 5*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 20*(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 - 10*(B*a^4*b + 2*A*a^3*b
^2)*d*e^5 + (B*a^5 + 5*A*a^4*b)*e^6)*x + 60*(3*B*b^5*d^6 - 2*(5*B*a*b^4 + A*b^5)*d^5*e + 6*(2*B*a^2*b^3 + A*a*
b^4)*d^4*e^2 - 6*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + (B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + (3*B*b^5*d^4*e^2 - 2*(5*B
*a*b^4 + A*b^5)*d^3*e^3 + 6*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 - 6*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + (B*a^4*b + 2*A
*a^3*b^2)*e^6)*x^2 + 2*(3*B*b^5*d^5*e - 2*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 6*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 - 6*
(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + (B*a^4*b + 2*A*a^3*b^2)*d*e^5)*x)*log(e*x + d))/(e^9*x^2 + 2*d*e^8*x + d^2*e
^7)

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giac [B]  time = 0.26, size = 887, normalized size = 2.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x, algorithm="giac")

[Out]

5*(3*B*b^5*d^4*sgn(b*x + a) - 10*B*a*b^4*d^3*e*sgn(b*x + a) - 2*A*b^5*d^3*e*sgn(b*x + a) + 12*B*a^2*b^3*d^2*e^
2*sgn(b*x + a) + 6*A*a*b^4*d^2*e^2*sgn(b*x + a) - 6*B*a^3*b^2*d*e^3*sgn(b*x + a) - 6*A*a^2*b^3*d*e^3*sgn(b*x +
a) + B*a^4*b*e^4*sgn(b*x + a) + 2*A*a^3*b^2*e^4*sgn(b*x + a))*e^(-7)*log(abs(x*e + d)) + 1/12*(3*B*b^5*x^4*e^
9*sgn(b*x + a) - 12*B*b^5*d*x^3*e^8*sgn(b*x + a) + 36*B*b^5*d^2*x^2*e^7*sgn(b*x + a) - 120*B*b^5*d^3*x*e^6*sgn
(b*x + a) + 20*B*a*b^4*x^3*e^9*sgn(b*x + a) + 4*A*b^5*x^3*e^9*sgn(b*x + a) - 90*B*a*b^4*d*x^2*e^8*sgn(b*x + a)
- 18*A*b^5*d*x^2*e^8*sgn(b*x + a) + 360*B*a*b^4*d^2*x*e^7*sgn(b*x + a) + 72*A*b^5*d^2*x*e^7*sgn(b*x + a) + 60
*B*a^2*b^3*x^2*e^9*sgn(b*x + a) + 30*A*a*b^4*x^2*e^9*sgn(b*x + a) - 360*B*a^2*b^3*d*x*e^8*sgn(b*x + a) - 180*A
*a*b^4*d*x*e^8*sgn(b*x + a) + 120*B*a^3*b^2*x*e^9*sgn(b*x + a) + 120*A*a^2*b^3*x*e^9*sgn(b*x + a))*e^(-12) + 1
/2*(11*B*b^5*d^6*sgn(b*x + a) - 45*B*a*b^4*d^5*e*sgn(b*x + a) - 9*A*b^5*d^5*e*sgn(b*x + a) + 70*B*a^2*b^3*d^4*
e^2*sgn(b*x + a) + 35*A*a*b^4*d^4*e^2*sgn(b*x + a) - 50*B*a^3*b^2*d^3*e^3*sgn(b*x + a) - 50*A*a^2*b^3*d^3*e^3*
sgn(b*x + a) + 15*B*a^4*b*d^2*e^4*sgn(b*x + a) + 30*A*a^3*b^2*d^2*e^4*sgn(b*x + a) - B*a^5*d*e^5*sgn(b*x + a)
- 5*A*a^4*b*d*e^5*sgn(b*x + a) - A*a^5*e^6*sgn(b*x + a) + 2*(6*B*b^5*d^5*e*sgn(b*x + a) - 25*B*a*b^4*d^4*e^2*s
gn(b*x + a) - 5*A*b^5*d^4*e^2*sgn(b*x + a) + 40*B*a^2*b^3*d^3*e^3*sgn(b*x + a) + 20*A*a*b^4*d^3*e^3*sgn(b*x +
a) - 30*B*a^3*b^2*d^2*e^4*sgn(b*x + a) - 30*A*a^2*b^3*d^2*e^4*sgn(b*x + a) + 10*B*a^4*b*d*e^5*sgn(b*x + a) + 2
0*A*a^3*b^2*d*e^5*sgn(b*x + a) - B*a^5*e^6*sgn(b*x + a) - 5*A*a^4*b*e^6*sgn(b*x + a))*x)*e^(-7)/(x*e + d)^2

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maple [B]  time = 0.07, size = 1205, normalized size = 2.84 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x)

[Out]

1/12*((b*x+a)^2)^(5/2)*(360*A*a*b^4*d^4*e^2*ln(e*x+d)+120*B*a^4*b*d*e^5*x-240*B*a^3*b^2*d^2*e^4*x+120*B*a^2*b^
3*d^3*e^3*x+60*B*a*b^4*d^4*e^2*x-54*A*b^5*d^5*e+720*B*a^2*b^3*d^4*e^2*ln(e*x+d)-600*B*a*b^4*d^5*e*ln(e*x+d)+21
0*A*a*b^4*d^4*e^2+90*B*a^4*b*d^2*e^4+180*A*a^3*b^2*d^2*e^4-300*A*a^2*b^3*d^3*e^3-6*B*a^5*d*e^5-300*B*a^3*b^2*d
^3*e^3+420*B*a^2*b^3*d^4*e^2-270*B*a*b^4*d^5*e+120*B*a^4*b*d*e^5*x*ln(e*x+d)-720*B*a^3*b^2*d^2*e^4*x*ln(e*x+d)
+1440*B*a^2*b^3*d^3*e^3*x*ln(e*x+d)-1200*B*a*b^4*d^4*e^2*x*ln(e*x+d)+240*A*a^2*b^3*d*e^5*x^2-330*A*a*b^4*d^2*e
^4*x^2-120*A*b^5*d^5*e*ln(e*x+d)-60*A*a^4*b*e^6*x+12*A*b^5*d^4*e^2*x+120*A*a^2*b^3*e^6*x^3+240*A*a^3*b^2*d*e^5
*x*ln(e*x+d)-720*A*a^2*b^3*d^2*e^4*x*ln(e*x+d)+720*A*a*b^4*d^3*e^3*x*ln(e*x+d)+60*B*a^4*b*d^2*e^4*ln(e*x+d)-36
0*B*a^3*b^2*d^3*e^3*ln(e*x+d)-240*A*b^5*d^4*e^2*x*ln(e*x+d)+360*B*b^5*d^5*e*x*ln(e*x+d)-6*A*a^5*e^6+66*B*b^5*d
^6+120*A*a^3*b^2*d^2*e^4*ln(e*x+d)-360*A*a^2*b^3*d^3*e^3*ln(e*x+d)-50*B*a*b^4*d*e^5*x^4-360*A*ln(e*x+d)*x^2*a^
2*b^3*d*e^5+360*A*ln(e*x+d)*x^2*a*b^4*d^2*e^4-360*B*ln(e*x+d)*x^2*a^3*b^2*d*e^5-30*A*a^4*b*d*e^5-240*B*a^2*b^3
*d*e^5*x^3+200*B*a*b^4*d^2*e^4*x^3+240*A*a^3*b^2*d*e^5*x-240*A*a^2*b^3*d^2*e^4*x+60*A*a*b^4*d^3*e^3*x+240*B*a^
3*b^2*d*e^5*x^2+40*A*b^5*d^2*e^4*x^3+120*B*a^3*b^2*e^6*x^3-60*B*b^5*d^3*e^3*x^3+126*A*b^5*d^3*e^3*x^2-204*B*b^
5*d^4*e^2*x^2+20*B*a*b^4*e^6*x^5+3*B*b^5*e^6*x^6+4*A*b^5*e^6*x^5-12*B*a^5*e^6*x+180*B*b^5*d^6*ln(e*x+d)-120*A*
a*b^4*d*e^5*x^3+120*A*ln(e*x+d)*x^2*a^3*b^2*e^6-120*A*ln(e*x+d)*x^2*b^5*d^3*e^3+60*B*ln(e*x+d)*x^2*a^4*b*e^6+1
80*B*ln(e*x+d)*x^2*b^5*d^4*e^2+630*B*a*b^4*d^3*e^3*x^2-660*B*a^2*b^3*d^2*e^4*x^2+720*B*ln(e*x+d)*x^2*a^2*b^3*d
^2*e^4-600*B*ln(e*x+d)*x^2*a*b^4*d^3*e^3-48*B*b^5*d^5*e*x-6*B*b^5*d*e^5*x^5+30*A*a*b^4*e^6*x^4-10*A*b^5*d*e^5*
x^4+60*B*a^2*b^3*e^6*x^4+15*B*b^5*d^2*e^4*x^4)/(b*x+a)^5/e^7/(e*x+d)^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^3,x)

[Out]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**3,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/(d + e*x)**3, x)

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